Place the chicken breasts in the bottom of the slow cooker. Add the drained black beans, diced tomatoes with green chilies, diced onion, and minced garlic on top.
Sprinkle the chili powder, cumin, and salt over the ingredients. Pour the chicken broth over everything.
Cover and cook on low for 6 to 7 hours, until the chicken reaches an internal temperature of 165°F and shreds easily.
Remove the chicken breasts, shred them with two forks, and return the meat to the slow cooker. Stir to combine.
Ladle the soup into bowls and top each serving with diced avocado and a lime wedge.